To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle 75° from the horizontal. An observer D = 565 m away measures the angle of elevation to the spot of light to be 45°. Find the height h of the cloud cover. 

This problem involves right triangle trigonometry, where the worker, the observer, and the spot of light form a triangle. The goal is to find the height \( h \) of the cloud cover.

### Let's break this down:
- The worker is shining a spotlight upward at an angle of \( 75^\circ \) from the horizontal.
- The observer is \( D = 565 \, \text{m} \) away from the worker.
- The observer measures the angle of elevation to the spot of light as \( 45^\circ \).

### Diagram and Variables:
- Let \( h \) be the height of the cloud cover.
- Let the point where the spotlight is located be point A.
- Let the observer be at point B, and the spot of light on the cloud cover be at point C.
- The triangle formed is a right triangle with:
  - Angle at \( A = 75^\circ \),
  - Angle at \( B = 45^\circ \).

We want to find the height \( h = BC \).

### Step-by-step solution:
Since the observer is directly below the cloud cover at point \( B \), the right triangle formed allows us to use trigonometric relationships to find the height.

The key idea is to use the tangent function, which relates the opposite and adjacent sides in a right triangle.

For angle \( B = 45^\circ \):

\[
\tan(45^\circ) = \frac{h}{D}
\]

Given that \( \tan(45^\circ) = 1 \), we can solve for \( h \):

\[
h = D \times \tan(45^\circ) = 565 \times 1 = 565 \, \text{m}
\]

Thus, the height of the cloud cover is:

\[
h = 565 \, \text{m}
\]

### Conclusion:
The height of the cloud cover is \( 565 \, \text{m} \).

Would you like further clarification, or do you have any additional questions?

Here are 5 related questions you might find interesting:
1. What if the observer measured a different angle, such as 60°? How would that change the calculation?
2. How would the situation change if the observer was closer to the worker, say 400 m away?
3. What trigonometric function would you use if you only knew the height and angle, and you wanted to find the distance between the observer and the worker?
4. How can you use the law of sines or cosines to solve for the unknowns in more complex triangles?
5. How would atmospheric conditions (such as refraction) affect the accuracy of this measurement?

**Tip:** In right-angle trigonometry, the tangent function is particularly useful when you know the angle and one of the sides adjacent or opposite the angle!

To measure the height of the cloud cover at an airport, a worker shines a spotlight upward at an angle 75° from the horizontal. An observer D = 565 m away measures the angle of elevation to the spot of light to be 45°. Find the height h of the cloud cover.

This problem demonstrates how to calculate the height of the cloud cover using trigonometry. A spotlight shines at an angle of 75°, and an observer 565 m away measures a 45° angle of elevation. Learn step-by-step how to apply the tangent function in right triangle trigonometry to find the cloud cover height. Ideal for students in Grades 9-11.

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